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19640Re: A point related to a quadrilateral

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  • jpehrmfr
    Jan 2, 2011
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      Dear Chris
      > > Consider a quadrilateral A_1,A_2,A_3,A_4
      > > For k =1,2,3,4, T_k is the triangle with vertices the A_i except A_k; O_k is the circumcenter of T_k, (O_k) the circumcircle and B_k the isogonal conjugate of A_k wrt T_k
      > > Then the inverse of B_k in (O_k) doesn't depend on k and this point M is the center of the homothecy mapping O_1,O_2,O_3,O_4 to B_1,B_2,B_3,B_4.
      > > Is there a special name for this point M? Do you know some references?
      > I do not know a special name for point M.
      > However I noticed this. Maybe you know it already.
      > Let T(A_1,A_2,A_3,A_4) = Transform A_1,A_2,A_3,A_4 --> O_1,O_2,O_3,O_4.
      > Then T^2(A_1,A_2,A_3,A_4) produces a quadrilateral homethetic with A_1,A_2,A_3,A_4 only rotated 180 degrees. Again Center of Homothecy = M.
      > T^4(A_1,A_2,A_3,A_4) produces a quadrilateral homothetic and with same orientation as A_1,A_2,A_3,A_4.

      Thank you for your nice remark.
      In fact, if O_1' is the circumcenter of O_2O_3O_4,...,
      the same homothecy maps A_i to O_i' and B_i to O_i
      The point M is characterized by the angular relations
      <A_iMA_j = <A_iA_kA_j +<A_iA_lA_j (oriented angles modulo Pi) where (i,j,k,l) is any permutation of (1,2,3,4)
      I would be very surprised if thete were no references about this configuration
      Friendly. Jean-Pierre
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